where “erfc” is the complementary error function, \(\delta _c\)  is the critical density contrast, and \(\sigma (M)\)  the variance of the density fluctuations at a mass scale \(M\) . When PBH formation takes place in a RD universe, \(\delta _c \simeq 0.414\)  [1]}. The variance of the density fluctuations at a mass scale \(M\)  is given in terms of the primordial spectrum as:
\(\sigma ^2(M)= \frac{4 (1+w)^2}{(5+3w)^2} \int d\ln k ~\left(\frac{k}{k_{M}}\right)^4 W^2[k/k_M] P_{\cal R}(k) \,,\) 
